Game Theory

1. The chairman of a prominent corporation has made unwelcome propositions and sexually harassed each of his 10 female employees. While these employees greatly dislike this treatment, each realizes that if she complains publicly, she may lose her job and her private life will be disrupted. If only one employee complains, she will not be believed and the chairman will super no consequences. If more employees complain, it becomes more likely that the employees will be believed and more likely that the chairman will be fired. If 5 or more employees complain, the chairman will surely be fired. There are two types of female employees. There are 3 Type A employees, who have very good prospects outside this corporation and there are 7 Type B employees whose careers would super badly if they lost this job. The cost of complaining for an employee depends on her type and on the total number who complain, as shown in the following table. In this table, when the total number of complaints is n, CA(n) is the cost of complaining for each Type A and CB(n) is the cost of complaining for each Type B. Table 1: Cost of Complaining

Numbers who complain	1	2	3	4	5 or more
CA(n): Cost to Type A	50	45	25	25	20
CB(n): Cost to Type B	100	90	60	55	40

The chances that the chairman will be fired increases as the number of complainers increases, up until there are 5 complaints, at which point the chairman is fired. All of the female employees hope that the chairman will be fired. The benefit to each female employee increases with the probability that the chairman is fired. If there are n complaints, the benefit to each female employee is B(n), as given in the table below. (Note that these benefits are received by every female employee, whether she complained or not.) If n employees complain, then the payoff to the employees who did not complain is B(n). The payoff to a type A employee who complained is B(n) CA(n). The payoff to a type B employee who complained is B(n) CB(n). Suppose that the employees must decide separately whether or not to complain. Thus each employee has only two possible strategies: Complain

Table 2: Benefit from complaints or not complain

n: Number of complaints	0	1	2	3	4	5 or more
B(n): Benefit to employees	0	0	10	40	50	100

A) Is there a Nash equilibrium in which nobody complains? Explain:

B) Are there any Nash equilibrium in which some, but not all, of the 10 female employees complain? If so, describe all of these equilibria.

C) Is there a Nash equilibrium in which all 10 female employees complain? Explain